Weakly dissipative semilinear equations of viscoelasticity

COMMUNICATIONS ON PURE AND APPLIED ANALYSIS Volume 0, Number 0, January 2005

Website: http://AIMsciences.org pp. 1–16

WEAKLY DISSIPATIVE SEMILINEAR EQUATIONS OF VISCOELASTICITY

Monica Conti Dipartimento di Matematica “F.Brioschi” Politecnico di Milano I-20133 Milano, Italy

Vittorino Pata Dipartimento di Matematica “F.Brioschi” Politecnico di Milano I-20133 Milano, Italy

(Communicated by Alain Miranville) Abstract. We consider an integro-partial differential equation of hyperbolic type with a cubic nonlinearity, in which no dissipation mechanism is present, except for the convolution term accounting for the past memory of the variable. Setting the equation in the history space framework, we prove the existence of a regular global attractor.

1. The Equation. Let Ω ⊂ R3 be a bounded domain with smooth boundary ∂Ω. For u : Ω × R → R, we consider the hyperbolic equation with memory on the time-interval R+ = (0, ∞) Z ∞ utt + αut − k(0)∆u − k 0 (s)∆u(t − s)ds + g(u) = f, (1) 0

arising in the theory of isothermal viscoelasticity (cf. [10, 22]). Here, α ≥ 0, g : R → R is a nonlinear term of (at most) cubic growth satisfying some dissipativity conditions, f : Ω → R is the external force, whereas the memory kernel k is a convex decreasing smooth function such that k(0) > k(∞) > 0. This equation is supplemented with the initial and boundary conditions   t ≤ 0, u(t) = u0 (t), (2) ut (0) = u1 ,   u(t)|∂Ω = 0, ∀t ∈ R, where the values u0 (t) (for t ≤ 0) and u1 are prescribed data. The terms contributing to dissipation in the above equation are αut (which is to be considered as a dynamical friction) and the convolution integral. Indeed, performing an integration by parts, it is easily seen that (1) can be transformed into Z ∞ £ ¤ utt + αut − k(∞)∆u − k(s) − k(∞) ∆ut (t − s)ds + g(u) = f. (3) 0

2000 Mathematics Subject Classification. 35B40, 35L70, 37L45, 45K05, 74D99. Key words and phrases. Hyperbolic equations with memory, dynamical systems, Lyapunov functionals, gradient systems, global attractors. 1

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M. CONTI AND V. PATA

Thus, in the limiting situation when k(s) − k(∞) is the Dirac mass at zero (possibly multiplied by a positive constant β), we obtain utt + αut − k(∞)∆u − β∆ut + g(u) = f, that is, the semilinear wave equation with weak damping and strong damping. This equation is well-known to generate a dynamical system on H01 (Ω)×L2 (Ω) possessing a global attractor of optimal regularity, provided that either α or β (or both) are strictly positive (see [1, 2, 11, 12, 15, 17, 20]). Setting µ(s) = −k 0 (s) (µ is positive and summable) and η t (s) = u(t) − u(t − s), equation (1) may be more conveniently rewritten as the system of equations on the time-interval R+ (cf. [8]) Z ∞  u + αu − k(∞)∆u − µ(s)∆η(s)ds + g(u) = f, tt t (4) 0  ηt = −ηs + ut . Accordingly, the initial and boundary conditions turn into   u(0) = u0 , ut (0) = u1 ,   0 η = η0 , where u0 = u0 (0) and η0 (s) = u0 (0) − u0 (−s), and, now for all t ≥ 0,   u(t)|∂Ω = 0, η t |∂Ω = 0,   t η (0) = 0.

(5)

(6)

The complete equivalence between the two formulations (1)-(2) and (4)-(6) is discussed in great detail in the review paper [14]. 2. Preliminary Discussion. As we will see in the next section, problem (4)-(6) generates a dissipative dynamical system on a suitable phase-space, usually called history space (cf. [14]). It is then of some interest to investigate the asymptotic properties of such a system, to see if the longterm dynamics is captured by (possibly small) subsets of the phase-space, such as bounded absorbing or attracting sets, or even global or exponential attractors. Of course, the stronger is the dissipation, the more are the chances that a satisfactory picture occurs. As pointed out before, here there are two terms that provide a favorable contribution: αut and R∞ − 0 µ(s)∆η(s)ds. As usual, good indications on the longterm behavior are given by the associated linear homogeneous system (i.e., when g and f both vanish), which has to be exponentially stable in order to have some hopes for the full system to possess (at least) a bounded absorbing set. This exponential stability holds true, provided that µ has an exponential decay at infinity. Most interesting, it holds true even if α = 0, namely, when all the dissipation is carried out by the memory. This result, proved in [9] exploiting techniques of linear semigroup theory, is far from being trivial, for in this case the dissipation is extremely weak, and the memory term is responsible of the uniform decay to zero of the whole associated energy. However, these methods, strictly related to the linear nature of the problem, cannot be exported to successfully attack the semilinear nonhomogeneous case. Clearly, if

SEMILINEAR EQUATIONS OF VISCOELASTICITY

3

α > 0, things are much easier, and the exponential decay can be proved via (quite) standard energy estimates. Going back to the full problem, existence of global and exponential attractors, along with regularity results, have been provided for α > 0 in [7, 5, 21]. A partial attempt to include the case α = 0 has been made in [13], where the existence of a global attractor (without any further regularity) is proved, but under quite strong assumptions (such as global Lipschitz continuity) on the nonlinearity g. The aim of this paper is to complete the analysis of the case α = 0, showing the existence of a global attractor of optimal regularity for a cubic nonlinearity. We want to emphasize straight away the main difficulty encountered here. The preliminary step to demonstrate the existence of global attractors for dynamical systems is the existence of bounded absorbing sets (usually obtained by means of energy estimates). For some reason, in the present case this seems out of reach. To overcome this (apparently) insurmountable obstacle, we shall make use of a general theorem (cf. [16, 18]), that allows to prove the existence of a global attractor without appealing to the existence of a bounded absorbing set, which is eventually obtained as a byproduct. The result applies to the so-called gradient systems, that is, dynamical systems possessing a global Lyapunov functional. Let us briefly describe the plan of this work. - In Section 3 we properly formulate the problem as a dynamical system on a suitable phase-space, and we prove the existence of a global Lyapunov functional. - In Section 4 we present the main result, whose proof is carried out in Sections 5,6,7. - In Section 8 we discuss some possible developments, whereas in Section 9 we reformulate our result in terms of trajectory attractors. - In the Appendix we recall the abstract theorem on the existence of global attractors for gradient systems. Notation. We denote by k · k and h·, ·i the norm and the inner product on L2 (Ω). Naming A = −∆ : D(A) = H01 (Ω) ∩ H 2 (Ω) ⊂ L2 (Ω) → L2 (Ω), we consider, for σ ∈ R, the Hilbert spaces H σ = D(Aσ/2 ), endowed with the standard inner products. Then, we introduce the L2 -weighted Hilbert spaces Mσ = L2µ (R+ , H σ+1 ), and the infinitesimal generator of the right-translation semigroup on M0 © ª T = −∂s : D(T ) = η(s) ∈ M0 : ηs ∈ M0 , η(0) = 0 ⊂ M0 → M0 , where ∂s is the distributional derivative with respect to the internal variable s. Finally, we define the product Hilbert spaces Hσ = H σ+1 × H σ × Mσ . Throughout the paper, c ≥ 0 will denote a generic constant, depending only on the structural parameters of the system under consideration (unless otherwise specified). Also, we shall often make use without explicit mention of the Young and the H¨older inequalities, as well as of the usual Sobolev embeddings. We conclude the section reporting a modified form of the Gronwall lemma, that we write in full generality for further references.

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M. CONTI AND V. PATA

Lemma 2.1. Let ψ : [0, ∞) → R be an absolutely continuous function, which fulfills for some ε > 0 and almost every t ≥ 0 the differential inequality d ψ(t) + 2εψ(t) ≤ h1 (t)ψ(t) + h2 (t), dt R t+1 Rt where τ h1 (y)dy ≤ m1 + ε(t − τ ), for all τ ∈ [0, t], and supt≥0 t |h2 (y)|dy ≤ m2 , for some constants m1 , m2 ≥ 0. Then there exist M1 , M2 ≥ 0 such that ψ(t) ≤ M1 |ψ(0)|e−εt + M2 ,

∀t ≥ 0.

Moreover, if m2 = 0 (that is, if h2 ≡ 0), it follows that M2 = 0. Proof. Fix t > 0. For τ ∈ [0, t], we set Z t ω(τ ) = h1 (y)dy − 2ε(t − τ ) ≤ m1 − ε(t − τ ). τ

Then, by the Gronwall lemma, Z t Z t ψ(t) ≤ ψ(0)eω(0) + eω(τ ) h2 (τ )dτ ≤ em1 |ψ(0)|e−εt + em1 e−εt eετ |h2 (τ )|dτ. 0

The inequality (cf. [19])

0

Z

t

eετ |h2 (τ )|dτ ≤

0

m2 eε eεt 1 − e−ε

yields the desired result. 3. The Associated Gradient System. We assume hereafter the following set of hypotheses. Let f ∈ H 0 be independent of time, and let g ∈ C 2 (R) satisfy the dissipation and the growth conditions |g 00 (y)| ≤ c(1 + |y|),

∀y ∈ R,

g(y) lim inf > −λ1 , y |y|→∞

(7) (8)

where λ1 is the first eigenvalue of A. Concerning the kernel µ(s) = −k 0 (s), we take µ ∈ W 1,1 (R+ ), µ ≥ 0, such that µ0 (s) + δµ(s) ≤ 0,

∀s ∈ R+ , for some δ > 0.

(9)

To avoid the presence of unnecessary constants, we assume the normalization condition Z ∞ µ(s)ds = 1, 0

and we put µ(0) = 1, which can always be done by rescaling µ, and changing δ accordingly. By the above assumptions, µ is decreasing (strictly, whenever µ(s) > 0) and µ(s) ≤ e−δs , ∀s ∈ R+ . Then, setting for simplicity k(∞) = 1, problem (4)-(6) for α = 0 can be translated into the evolution equation in H0  Z ∞   u + Au + µ(s)Aη(s)ds + g(u) = f, t ∈ R+ ,  tt  0

ηt = T η + u t , t ∈ R+ ,     (u(0), ut (0), η 0 ) = (u0 , u1 , η0 ).

(10)

SEMILINEAR EQUATIONS OF VISCOELASTICITY

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This equation (cf. [7, 21] and references therein) generates a C0 -semigroup or dynamical system S(t) on the phase-space H0 . In particular, the third component η of the solution has the explicit representation formula ( u(t) − u(t − s), 0 < s ≤ t, t η (s) = (11) η0 (s − t) + u(t) − u0 , s > t. Note that the set S of stationary points of S(t) is made of vectors (u0 , 0, 0) ∈ H0 , with u0 solution to the elliptic problem Au0 + g(u0 ) = f. Appealing to (8), it is then clear that S is bounded in H0 . According to Definition A.1 in the Appendix, we have Proposition 3.1. S(t) is a gradient system on H0 . Proof. We define the function Φ ∈ C(H0 , R) as Φ(u, v, η) = E(u, v, η) + G(u) − hf, ui, where E(u, v, η) =

¢ 1 ¡ 1/2 2 kA uk + kvk2 + kηk2M0 2

is the energy functional, and Z Z

u(x)

G(u) =

g(y)dydx. Ω

0

On account of (7)-(8), it is readily seen that c0 E − c ≤ Φ ≤ cE 2 + c, for some c0 > 0, possibly very small, which depends only on the value of the limit in (8). We now show that Φ is decreasing along the trajectories of S(t). Indeed, multiplying the first equation of (10) by ut in H 0 and the second by η in M0 , we get Z d 1 ∞ 0 Φ= µ (s)kA1/2 η(s)k2 ds ≤ 0. (12) dt 2 0 Finally, if Φ is constant along a trajectory of S(t), using (9) we learn that kη t kM0 = 0 for all t ≥ 0. By force of (11), we conclude that (u(t), ut (t), η t ) = (u0 , 0, 0) ∈ S. Hence, Φ is a Lyapunov functional. 4. The Global Attractor. Our main result reads as follows. Theorem 4.1. The dynamical system S(t) on H0 generated by (10) possesses a (unique) connected global attractor A. Moreover, A is contained and bounded in H1 . The existence of the global attractor A will be obtained by applying Theorem A.3. Indeed, in light of Proposition 3.1 and the boundedness of the set S, it will suffice to find, in correspondence to any given bounded set B ⊂ H0 , a decomposition S(t) = L(t) + N (t) as in Remark A.5. The regularity of A will be proved via a bootstrap argument envisaged in [15].

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M. CONTI AND V. PATA

In order to accomplish this program, we write (cf. [15]) g = g0 + g1 , where g0 , g1 ∈ C 2 (R) fulfill |g000 (y)| ≤ c(1 + |y|), g00 (0)

∀y ∈ R,

= 0,

(13) (14)

g0 (y)y ≥ 0,

∀y ∈ R,

(15)

|g10 (y)|

∀y ∈ R.

(16)

≤ c,

Then, we fix a bounded set B ⊂ H0 and, for z = (u0 , u1 , η0 ) ∈ B, we split the solution S(t)z = (u(t), ut (t), η t ) into the sum L(t)z + N (t)z, where L(t)z = (v(t), vt (t), ξ t )

and

N (t)z = (w(t), wt (t), ζ t )

solve the problems Z ∞    v + Av + µ(s)Aξ(s)ds + g0 (v) = 0,   tt 0 ξt = T ξ + vt ,     (v(0), vt (0), ξ 0 ) = z, and

Z    w + Aw + tt  

(17)

∞

µ(s)Aζ(s)ds + g(u) − g0 (v) = f,

0

(18)

ζt = T ζ + wt ,     (w(0), wt (0), ζ 0 ) = 0.

The next sections will be devoted to prove the uniform decay of L(t) and the smoothing property of N (t). 5. Some Preliminary Lemmata. We begin to establish some estimates that shall be needed in the course of the investigation. Here, as in the sequel, we shall perform formal multiplications, which are justified within a proper regularization scheme. Lemma 5.1. Let σ ∈ [0, 1] be given, and let z ∈ Hσ such that kzkH0 ≤ R for some R ≥ 0. Assuming that S(t)z ∈ Hσ for all t > 0, there exist two constants m, mR > 0 (the first independent of R) such that the inequality −

d 1 mR hut , ηiMσ−1 ≤ εkA(σ+1)/2 uk2 − kAσ/2 ut k2 + kηk2Mσ dt 2 ε Z ∞

−m

µ0 (s)kA(σ+1)/2 η(s)k2 ds + ε

0

holds for all ε ∈ (0, 1). The same estimate is valid replacing (u, ut , η) with (w, wt , ζ) and, for σ = 0, with (v, vt , ξ). In this latter case, without the last term ε appearing in the right-hand side. Proof. We write the proof for (u, ut , η) (the other two cases are treated similarly). Using the second equation of (10), we have −

d hut , ηiMσ−1 = −kAσ/2 ut k2 − hut , T ηiMσ−1 − hutt , ηiMσ−1 . dt

SEMILINEAR EQUATIONS OF VISCOELASTICITY

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Integrating by parts with respect to s, in light of the decay of µ and of the equality η(0) = 0, we obtain Z ∞ −hut , T ηiMσ−1 = − µ0 (s)hAσ/2 ut , Aσ/2 η(s)ids 0 ³ Z ∞ ´1/2 σ/2 ≤ kA ut k − µ0 (s)kAσ/2 η(s)k2 ds 0 Z ∞ 1 σ/2 2 µ0 (s)kA(σ+1)/2 η(s)k2 ds, ≤ kA ut k − m 2 0 1 2λ1 .

for m =

Concerning the other term, the first equation of (10) yields °Z ∞ °2 ° ° −hutt , ηiMσ−1 = hu, ηiMσ + ° µ(s)A(σ+1)/2 η(s)ds° + hg(u) − f, ηiMσ−1 . 0

But

Z (σ+1)/2

hu, ηiMσ ≤ kA

uk

∞

µ(s)kA(σ+1)/2 η(s)kds

0

≤ kA(σ+1)/2 ukkηkMσ 1 ≤ εkA(σ+1)/2 uk2 + kηk2Mσ , 4ε and

°Z ° °

∞

µ(s)A

(σ+1)/2

°2 ³ Z ° η(s)ds° ≤

0

∞

µ(s)kA(σ+1)/2 η(s)kds

0

´2

≤ kηk2Mσ .

Finally, since σ ≤ 1 and kA1/2 uk is bounded (with a bound depending on R), Z ∞ cR hg(u) − f, ηiMσ−1 ≤ kg(u) − f k µ(s)kAσ η(s)kds ≤ kηk2Mσ + ε, ε 0 for some cR > 0. Note that, to prove the analogous result for (v, vt , ξ), this term becomes hg0 (v), ξiM−1 , which, using the fact that g0 (0) = 0, is controlled by cR hg0 (v), ξiM−1 ≤ cR kA1/2 vkkξkM0 ≤ εkA1/2 vk2 + kξk2M0 . ε Setting mR = cR + 45 , we conclude that mR kηk2Mσ + ε. ε Collecting the two above estimates, we reach the desired conclusion. −hutt , ηiMσ−1 ≤ εkA(σ+1)/2 uk2 +

Lemma 5.2. Assume that kzkH0 ≤ R for some R ≥ 0. Then, for every ε > 0, there 0 exists Mε,R > 0 such that Z t 0 kut (y)k2 dy ≤ ε(t − τ ) + Mε,R , τ

for every t ≥ τ ≥ 0. Proof. In this proof, the generic constant c may depend on R. Define I(t) = 2Φ(t) − 2ε2 hut (t), η t iM−1 ,

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M. CONTI AND V. PATA

where Φ is the Lyapunov functional. Notice that I is bounded (with a bound depending on R). Then, collecting (12) and Lemma 5.1, and exploiting (9), ¡ ¢ d I + ε2 kut k2 ≤ − δ − 2mδε2 − 2mR ε kηk2M0 + cε3 ≤ cε3 , dt for ε small enough. In that case, an integration on (τ, t) yields the required inequality, which then clearly holds for every ε > 0. Lemma 5.3. Let σ ∈ [0, 1] be given, and assume that kzkHσ ≤ R for some R ≥ 0. σ Then there exists CR ≥ 0 such that σ , kS(t)zkHσ ≤ CR

∀t ≥ 0.

σ ≥ 0 such that Moreover, for all ε > 0 there exists Mε,R Z t σ kAσ/2 ut (y)k2 dy ≤ ε(t − τ ) + Mε,R , τ

for every t ≥ τ ≥ 0. Proof. The result for σ = 0 is already known. Therefore, consider σ ∈ (0, 1]. In the following, the generic constant c may depend on R (in fact, c will depend only on the H0 -norm of z). For ε ∈ (0, 1) and ν ≥ 0, we define the functional Jσ (t) = Eσ (t) + Fσ (t) − ε2 hut (t), η t iMσ−1 + νε2 hAσ/2 ut (t), Aσ/2 u(t)i, where

¢ 1 ¡ (σ+1)/2 kA u(t)k2 + kAσ/2 ut (t)k2 + kη t k2Mσ 2 is the higher-order energy functional, and Eσ (t) =

Fσ (t) = hg(u(t)), Aσ u(t)i − hf, Aσ u(t)i. It is clear that, for ε small enough, 1 Eσ (t) − c ≤ Jσ (t) ≤ 2Eσ (t) + c. 2 Also, using the first equation of (10), and recalling the embeddings H (2+σ)/2 ,→ L6/(1=σ) (Ω)

and

H 1−σ ,→ L6/(1+2σ) (Ω),

it is straightforward to check that Z ¢ 1 ∞ 0 d¡ Eσ + Fσ − µ (s)kA(σ+1)/2 η(s)k2 ds dt 2 0 = hg 0 (u)ut , Aσ ui ¡ ¢ ≤ c 1 + kuk2L6/(1−σ) kut kkAσ ukL6/(1+2σ) ¡ ¢ ≤ c 1 + kA1/2 ukkA(1+σ)/2 uk kut kkA(1+σ)/2 uk ≤ ckut k + ckut kEσ , and 1 d σ/2 hA ut , Aσ/2 ui ≤ kAσ/2 ut k2 − kA(σ+1)/2 uk2 + kηk2Mσ + c. dt 2

(19)

SEMILINEAR EQUATIONS OF VISCOELASTICITY

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Exploiting now (9) and Lemma 5.1, and collecting the two above inequalities, we end up with ³ν ´ ³1 ´ d Jσ + ε2 − ε kA(σ+1)/2 uk2 + ε2 − ν kAσ/2 ut k2 (20) dt 2 2 ³δ ´ + − mδε2 − νε2 − mR ε kηk2Mσ 2 ≤ ckut k + ckut kEσ + cνε2 + ε3 . Choosing now ν = turns into

1 4

and ε small enough, in light of (19), the above inequality

d Jσ + ε0 Jσ ≤ c + ckut k + ckut kJσ , dt for some ε0 > 0. Then, from Lemma 5.2 and the generalized Gronwall Lemma 2.1 we get the first part of the thesis. To obtain the integral control, we consider again (20) with ν = 0 and ε sufficiently small. Notice that now we know that Eσ ≤ c. Therefore we are led to d ε2 Jσ + kAσ/2 ut k2 ≤ ckut k + ε3 . dt 2 An integration on (τ, t), together with a further application of Lemma 5.2, complete the argument. 6. Exponential Decay of L(t). We now prove that the C0 -semigroup L(t) on H0 has an exponential decay. Lemma 6.1. There exists κ > 0 and an increasing positive function Γ such that kL(t)zkH0 ≤ Γ(R)e−κt ,

∀t ≥ 0,

whenever kzkH0 ≤ R. Proof. In this proof, the generic constant c may depend on R. Let us consider the functional Z Z v(x,t) ε2 J(t) = E(v(t), vt (t), ξ t ) + g0 (y)dydx − ε2 hvt (t), ξ t iM−1 + hvt (t), v(t)i. 4 Ω 0 By virtue of (15), if ε is small enough there holds 1 E(v(t), vt (t), ξ t ) ≤ J(t) ≤ cE(v(t), vt (t), ξ t ). 2 Thus, on account of Lemma 5.1, and provided that ε is small enough, with calculations similar to those in the proof of Lemma 5.3, we obtain the inequality d J + 2κR J ≤ 0, dt for some κR > 0, and the Gronwall Lemma yields kL(t)zkH0 ≤ ce−κR t ,

∀t ≥ 0.

It is then standard matter to see that κR can be replaced by, say, κ1 (that is, a positive constant independent of R), upon increasing the above constant c accordingly.

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M. CONTI AND V. PATA

Incidentally, if in the original system g1 = 0 and f = 0, then S(t) = L(t). In that case, we have proved the exponential stability of the whole semigroup S(t). In fact, the result can be generalized to the equation, recently considered in [3], Z ∞ m utt + α|ut | ut + k(0)Au + k 0 (s)Au(t − s)ds + g0 (u) = 0, 0

which, setting k(∞) = 1, translates in the history space setting into the system Z ∞  u + α|u |m u + Au + µ(s)Aη(s)ds + g0 (u) = 0, tt t t (21) 0  ηt = T η + ut . Then we have Proposition 6.1. Let m ∈ [0, 4]. If α ∈ L6/(4−m) (Ω), and α ≥ 0, then the energy associated to (21) exponentially decays to zero. Proof. We limit ourselves to give a brief outline of the argument, which parallels the one of Lemma 6.1, with minor modifications. First, the solution u is easily seen to belong to the space Lm+2 (R+ , Lm+2 (Ω)). As before, we consider the functional α J(t) (now for (u, ut , η)). The time derivative of J produces in the left-hand side of the differential inequality the extra term kut km+2 + ε2 hα|ut |m ut , ηiM−1 − Lm+2 α

For any q ∈

H01 (Ω),

ε2 hα|ut |m ut , ui. 4

there holds 1/(m+2)

|hα|ut |m ut , qi| ≤ kαkL6/(4−m) kut km+1 kqkL6 ≤ ckut km+1 kA1/2 qk. Lm+2 Lm+2 α

α

Hence, we conclude that for every ρ > 0 there is cρ > 0 such that ¯ ¯ ¡ 1/2 ¢m+2 1 ¯ ¯ uk + kηkM0 . ¯hα|ut |m ut , ηiM−1 − hα|ut |m ut , ui¯ ≤ cρ kut km+2 m+2 + ρ kA L α 4 At this stage, we already know that the energy is bounded, with a bound depending on the norm of the initial data, thus, the extra term is greater than or equal to ¡ ¢ (1 − cρ ε2 )kut km+2 − cρε2 kA1/2 uk2 + kηk2M0 . Lm+2 α

Therefore, fixing ρ small enough, the second term is controlled by the analogous one appearing in the inequality, whereas the first one can be cancelled, provided we subsequently choose ε small enough. In conclusion, we recover d J + 2κR J ≤ 0, dt which entails the result. Remark 6.1. Proposition 6.1 extends the main result of [3]. In particular, it is worth noting that α can be unbounded (for m < 4), so that, in some zones of the domain Ω, the damping can be extremely effective. This, in conformity with the physical viewpoint, plays against dissipativity (think, for instance, to the simple situation of the damped pendulum). Also, observe that in in [3] the initial history is null, that is, the convolution integral is taken from 0 to t, whereas the result holds true for nontrivial initial histories as well.

SEMILINEAR EQUATIONS OF VISCOELASTICITY

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7. Compactness of N (t). We finally proceed to establish the compactness property of N (t). First, we prove Lemma 7.1. Let σ ∈ [0, 34 ] be fixed, and assume that kzkHσ ≤ R for some R ≥ 0. σ Then there exists KR ≥ 0 such that σ kN (t)zkH(1+4σ)/4 ≤ KR ,

∀t ≥ 0.

Proof. Here, the generic constant c may depend on R. The proof strictly follows the arguments of Lemma 5.3, with N (t) in place of S(t) (working with the exponent σ + 41 ). Hence, for ε ∈ (0, 1), we define the functional JσN (t) = EσN (t)+FσN (t)−ε2 hwt (t), ζ t iM(4σ−3)/4 +

ε2 (1+4σ)/8 hA wt (t), A(1+4σ)/8 w(t)i, 4

where

¢ 1 ¡ (5+4σ)/8 kA w(t)k2 + kA(1+4σ)/8 wt (t)k2 + kζ t k2M(1+4σ)/4 2 is the energy of order σ + 41 of (w(t), wt (t), ζ t ), and EσN (t) =

FσN (t) = hg(u(t)) − g0 (v(t)), A(1+4σ)/4 w(t)i − hf, A(1+4σ)/4 w(t)i. Again, for ε small enough, 1 N E (t) − c ≤ JσN (t) ≤ 2EσN (t) + c. 2 σ On account of the first equation of (18), we get Z ¢ 1 ∞ 0 d¡ N N E + Fσ − µε (s)kA(5+4σ)/8 ζ(s)k2 ds dt σ 2 0 = h(g00 (u) − g00 (v))ut , A(1+4σ)/4 wi + hg00 (v)wt , A(1+4σ)/4 wi + hg10 (u)ut , A(1+4σ)/4 wi ε2 (1+4σ)/8 kA wt k2 + ckAσ/2 ut k2 + ckAσ/2 ut kEσN + ckA1/2 vkEσN + c. 8 The last inequality follows from Sobolev embeddings (cf. [7, 15], where analogous computations are encountered). At this stage, the above constant c depends on ε, which, however, will be eventually fixed. Indeed, on account of Lemma 5.1, choosing ε small enough, we are led to d N J + 2ε0 JσN ≤ c + h + hJσN , dt σ for some ε0 > 0, where ≤

h(t) = ckA1/2 v(t)k + ckAσ/2 ut (t)k. Notice that from Lemma 5.3 and Lemma 6.1 we have Z t h(y)dy ≤ ε0 (t − τ ) + c. τ

Hence, the thesis follows by applying the generalized Gronwall Lemma 2.1. We are now in a position to demonstrate the existence of the global attractor. This will follow from the next lemma. Lemma 7.2. There exists a compact set K ⊂ H0 such that N (t)B ⊂ K,

∀t ≥ 0.

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Proof. Let R = supz∈B kzkH0 . From Lemma 7.1, if z ∈ B then 0 kN (t)zkH1/4 ≤ KR ,

∀t ≥ 0.

The representation formula (11) for ζ reads ( w(t) − w(t − s), 0 < s ≤ t, t ζ (s) = w(t), s > t. Thus, as ζ t (0) = 0,

( wt (t − s), 0 < s ≤ t, ∂s ζ (s) = 0, s > t. t

Since the kernel µ is decreasing, it is then apparent that ζ t is bounded in M1/4 and ∂s ζ t is bounded in M−3/4 , uniformly as z ∈ B and t ≥ 0. Moreover sup sup sup kA1/2 ζ t (s)k < ∞. t≥0 s∈R+ z∈B

Hence, from the compactness lemma [21, Lemma 5.5], ζ t lies in a compact subset of M0 . Since (w(t), wt (t)) is bounded in H 5/4 × H 1/4 b H 1 × H 0 , we are done. Summarizing what we have seen so far, the dynamical system S(t) on H0 fulfills the hypotheses of the abstract Theorem A.3. Therefore, there exists the global attractor A. Moreover (see Remark A.5) A is a bounded subset of H1/4 . To complete the proof of Theorem 4.1, we are left to show that A is a bounded subset of H1 . But if z ∈ A, exploiting the full invariance property of the global attractor, we apply again Lemma 6.1 and Lemma 7.1, for data z ∈ A, and we discover that in fact A is bounded in H1/2 . Repeating twice this bootstrap argument (cf. [15]), we end up to prove the desired boundedness in the space H1 . 8. Further Developments. As remarked at the beginning, the main difficulty here was to find a compact attracting set without having at our disposal a bounded absorbing set. Once this obstacle has been removed, the asymptotic analysis can be pushed much further the sole existence of the global attractor. As a matter of fact, we can even consider, in the spirit of [6, 7], a one-parameter family of equations. Namely, Z ∞  u + Au + µ (s)Aη(s)ds + g(u) = f, tt

ε

0  ηt = T η + u t ,

where, for ε ∈ (0, 1],

1 ³s´ µ . ε2 ε The limiting situation ε = 0 corresponds to the strongly damped wave equation µε (s) =

utt + Au + Aut + g(u) = f. Then, for every ε ∈ [0, 1], we have a dynamical system Sε (t) acting on the phasespace ( H 1 × H 0 × L2µε (R+ , H 1 ), ε > 0, 0 Hε = H 1 × H 0, ε = 0. Following the ideas of [6, 7], it is possible to prove the existence of a family Eε of exponential attractors for Sε (t) of (uniformly) bounded fractal dimension, which is stable for the singular limit ε → 0, with respect to the symmetric Hausdorff

SEMILINEAR EQUATIONS OF VISCOELASTICITY

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distance. In particular, this means that the global attractor A of the original problem has finite fractal dimension. Also, requiring additional regularity on f and g, along with some compatibility conditions for g, A can be proved to be a bounded subset of Hm , with m ∈ N as big as f and g permit (cf. [6]). We will possibly discuss these issues in more detail in forthcoming papers. 9. An Alternative Point of View. V.V. Chepyzhov and A. Miranville considered in [5] equation (1), for α > 0. They introduced the space © ª E − = (u(t), ut (t)) : u ∈ Cb (R− , H 1 ), ut ∈ Cb (R− , H 0 ) , where R− = (−∞, 0] and Cb stands for continuous and bounded, and defined the C0 -semigroup Σ(τ ) acting on E − by the formula (Σ(τ )u0 )(t) = u(t + τ )|t≤0 , where u(t) is the solution to (1) with initial data u0 ∈ E − . Then, they proved Theorem 9.1. Σ(τ ) has a (unique) connected global attractor A− ⊂ E − . Moreover, A− is bounded in Cb (R− , H 1 × H 2 ). In their terminology, the global attractor A− is a compact set in Cloc (R− , H 1 × H ) (i.e., the space of continuous functions endowed with the local uniform convergence topology), bounded in E − , strictly invariant under the action of Σ(τ ), which attracts any bounded set B− ⊂ E − in the topology of Cloc (R− , H 1 × H 0 ), namely, for every T > 0, h i lim distC([−T,0],H 1 ×H 0 ) (Σ(τ )B− , A− ) = 0, 0

τ →∞

where “dist” denotes the usual Hausdorff semidistance. This approach, in some sense more natural, can actually be recovered within the history framework. Indeed, the recent paper [4] proves (for a wide class of equations, including this one) that if the associated semigroup S(t) acting on the history phase-space H0 possesses a global attractor A, then Σ(τ ) possesses a global attractor A− , given by © ª A− = (u(t), η t ), t ∈ R− : (u(0), η 0 ) ∈ A . Hence, in light of our results, and exploiting the full invariance of the attractor, we have Theorem 9.2. The semigroup Σ(τ ) corresponding to (1) with α = 0 possesses a (unique) connected global attractor (in the sense of Chepyzhov-Miranville) A− ⊂ E − . In addition, A− is bounded in L∞ (R− , H 2 × H 1 ). Remark 9.1. In fact, since S(t) is a dynamical system also on the phase-space H1 , A− is bounded in Cb (R− , H 2 × H 1 ). Appendix: The Abstract Result. Let S(t) : X → X be a C0 -semigroup on a Banach space X . Definition A.1. A function Φ ∈ C(X , R) is called a Lyapunov functional if - Φ(u) → ∞ if and only if kukX → ∞; - Φ(S(t)x) is nonincreasing for any x ∈ X ; - if Φ(S(t)x) = Φ(x) for all t > 0, then x is a stationary point for S(t). If there exists a Lyapunov functional, then S(t) is called a gradient system.

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Remark A.2. The existence of a Lyapunov functional ensures that bounded sets have bounded orbits. Recall that the Kuratowski measure of noncompactness α(B) of a bounded set B ⊂ X is defined by © ª α(B) = inf d : B has a finite cover of balls of X of diameter less than d . We now state the result, which can be found in [16, 18]. However, for the reader convenience, we include here a self-contained rather simple proof. Theorem A.3. Assume the following conditions. (i) There exists a Lyapunov functional Φ. (ii) The set S of stationary points is bounded in X . (iii) For every nonempty bounded set B ⊂ X we have that lim α(S(t)B) = 0. t→∞

Then S(t) possesses a connected global attractor A that consists of the unstable manifold of the set S. Proof. Let B ⊂ X be a nonempty bounded set. Then, on account of (iii), the ωlimit set ω(B) is nonempty, compact, positively invariant, and attracts B. Let C be the bounded and positively invariant set defined by n o C = x ∈ X : Φ(x) < sup Φ(u) + 1 . u∈S ∗

∗

∗ Observe that there exists ¡ ¢ t = t (B) ≥ 0 such that S(t )ω(B) ⊂ C. Indeed, for every x ∈ X , distX S(t)x, S → 0, and by the continuity of S(t), for every k ∈ ω(B) there exists a neighborhood Uk of k and a time tk > 0 such that S(t)Uk ⊂ C, for every ©t ≥ tk . Extract then a finite subcover Uk1 , . . . , Ukn of ω(B), and set ª t∗ = max tk1 , . . . , tkn . The attraction property of ω(B) implies the existence of a positive function ψ(t) vanishing at infinity such that, for every x ∈ B, we can write S(t)x = k(t) + q(t), with k(t) ∈ ω(B) and kq(t)kX ≤ ψ(t). Thus, for a given x ∈ B, we have £ ¤ S(t∗ + t)x = S(t∗ )k(t) + S(t∗ ) k(t) + q(t) − S(t∗ )k(t). Notice that S(t∗ )k(t) ∈ S(t∗ )ω(B) ⊂ C. Moreover, the continuity S(t∗ ) ∈ C(X , X ) implies the uniform continuity of S(t∗ ) in a neighborhood of the compact set ω(B). Thus, £ ¤ kS(t∗ ) k(t) + q(t) − S(t∗ )k(t)kX ≤ 1, provided that ψ(t) is small enough, that is to say, provided that t is large enough. We conclude that the ball B0 of X centered at zero of radius R0 = supv∈C kvkX + 1 is an absorbing set for S(t). Then, applying (iii) with B0 in place of the generic bounded set B, the thesis follows by standard arguments of the theory of dynamical systems.

Remark A.4. In [16], Φ is required to be bounded below. However, this condition does not seem to be necessary. On the contrary, the requirement that if Φ(u) → ∞ then kukX → ∞ (there overlooked) is crucial to guarantee the boundedness of orbits of bounded sets. Remark A.5. Condition (iii) of the theorem is usually verified by finding, for any given bounded set B ⊂ X , a decomposition S(t) = L(t) + N (t) such that h i lim sup kL(t)xkX = 0, t→∞

x∈B

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and N (t)B ⊂ K(t), where K(t) is a compact set depending on t and B. It can also happen that K(t) = K (i.e, independent of t). In that case, the global attractor A belongs to the compact set K associated with B0 . Acknowledgments. We thank the referee for careful reading and valuable comments. REFERENCES [1] J. Arrieta, A.N. Carvalho, J.K. Hale, A damped hyperbolic equation with critical exponent, Commun. Partial Differential Equations, 17 (1992), 841–866. [2] A.V. Babin, M.I. Vishik, Regular attractors of semigroups and evolution equation, J. Math. Pures Appl., 62 (1983), 441–491. [3] S. Berrimi, S.A. Messaoudi, Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping, Electron. J. Differential Equations, no.88 (2004), 10 pp. [4] V.V. Chepyzhov, S. Gatti, M. Grasselli, A. Miranville, V. Pata, Trajectory and global attractors for evolution equations with memory, submitted. [5] V.V. Chepyzhov, A. Miranville, Trajectory and global attractors of dissipative hyperbolic equations with memory, Commun. Pure Appl. Anal., to appear. [6] M. Conti, V. Pata, M. Squassina, Singular limit of differential systems with memory, Indiana Univ. Math. J., to appear. [7] M. Conti, V. Pata, M. Squassina, Singular limit of dissipative hyperbolic equations with memory, submitted. [8] C.M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297–308. [9] M. Fabrizio, B. Lazzari, On the existence and asymptotic stability of solutions for linear viscoelastic solids, Arch. Rational Mech. Anal., 116 (1991), 139–152. [10] M. Fabrizio, A. Morro, “Mathematical problems in linear viscoelasticity”, SIAM Studies Appl. Math. 12, SIAM, Philadelphia, 1992. [11] J.M. Ghidaglia, A. Marzocchi, Longtime behaviour of strongly damped wave equations, global attractors and their dimension, SIAM J. Math. Anal., 22 (1991), 879–895. [12] J.M. Ghidaglia, R. Temam, Attractors for damped nonlinear hyperbolic equations, J. Math. Pures Appl., 66 (1987), 273–319. [13] C. Giorgi, J.E. Mu˜ noz Rivera, V. Pata, Global attractors for a semilinear hyperbolic equation in viscoelasticity, J. Math. Anal. Appl., 260 (2001), 83–99. [14] M. Grasselli, V. Pata, Uniform attractors of nonautonomous systems with memory, in “Evolution Equations, Semigroups and Functional Analysis” (A. Lorenzi and B. Ruf, Eds.), pp.155–178, Progr. Nonlinear Differential Equations Appl. no.50, Birkh¨ auser, Boston, 2002. [15] M. Grasselli, V. Pata, Asymptotic behavior of a parabolic-hyperbolic system, Commun. Pure Appl. Anal., 3 (2004) 849–881. [16] J.K. Hale, “Asymptotic behavior of dissipative systems”, Amer. Math. Soc., Providence, 1988. [17] A. Haraux, Two remarks on dissipative hyperbolic problems, in “S´ eminaire du Coll` ege de France”, J.-L. Lions Ed., Pitman, Boston, 1985. [18] O.A. Ladyzhenskaya, Finding minimal global attractors for the Navier-Stokes equations and other partial differential equations, Russian Math. Surveys, 42 (1987), 27– 73. [19] V. Pata, G. Prouse, M.I. Vishik, Traveling waves of dissipative non-autonomous hyperbolic equations in a strip, Adv. Differential Equations, 3 (1998), 249–270. [20] V. Pata, M. Squassina, On the strongly damped wave equation, Comm. Math. Phys., to appear. [21] V. Pata, A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505–529.

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Received December 2004; revised January 2005. E-mail address: [email protected] E-mail address: [email protected]